+343 If V={(x,y)|x,y ∈ R} with acts (x1,y1)@(x2,y2)=\((\sqrt[3]{(x1^3+x2^3)},\sqrt[3]{(y1^3+y2^3)})\) and a$(x1,y1)=(ax1,ay1) , a ∈ R.
Prove V with this acts is NOT vector space.Thank you!
+6250 are you sure it's not?
it's addition has commutativity and associativity
(0,0) is the clear additive identity and (-x,-y) is the additive inverse of (x,y)
the scalar properties are identical to the Euclidean vector space
I'm not seeing why this isn't a vector space.
remember the cube and cube root form a bijection
This is not a vector space. Exactly one of the axioms of a vector space is not satisfied.
+343 Rom yes im sure the exercise say it's not.For this reason i can't solve the exercise
+343 Guest "Exactly one of the axioms of a vector space is not satisfied"
Witch is this axiom?
+6250 Ok. Distribution of scalar mutliplication with respect to field addition doesn't hold.
we should have
\((a+b)\vec{v} = a \vec{v} \oplus b \vec{v}\)
let's take a look
\((a+b)\vec{v} =\{(a+b)v_x,(a+b)v_y\}\\ a\vec{v} \oplus b \vec{v} =\left \{\sqrt[3]{a^3 v_x^3+b^3 v_x^3}, \sqrt[3]{a^3 v_y^3+b^3 v_y^3 }\right \}=\\ \left\{\sqrt[3]{a^3+b^3}v_x,~\sqrt[3]{a^3+b^3}v_y \right\}\\ \sqrt[3]{a^3+b^3} \neq a+b\)
